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NeuroQuantology | June 2013 | Volume 11 | Issue 2 | Page 197-208

Beauvais F., Benveniste’s experiments and quantum-li ke probabilities

www.neuroquantology.com

197

Quantum-Like Interferences of Experimenter's

Mental States: Application to “Paradoxical”

Results in Physiology

Francis Beauvais

ABSTRACT

Objectives: “Memory of water” experiments (also known as Benveniste’s experiments) were the source of a

famous controversy in the contemporary history of sciences. We recently proposed a formal framework devoid of

any reference to “memory of water” to describe these disputed experiments. In this framework, the results of

Benveniste’s experiments are seen as the consequence of quantum-like interferences of cognitive states. Design:

In the present article, we describe retrospectively a series of experiments in physiology (Langendorff preparation)

performed in 1993 by Benveniste’s team for a public demonstration. These experiments aimed at demonstrating

“electronic transmission of molecular information” from protein solution (ovalbumin) to naïve water. The

experiments were closely controlled and blinded by participants not belonging to Benveniste’s team. Results: The

number of samples associated with signal (change of coronary flow of isolated rodent heart) was as expected; this

was an essential result since, according to mainstream science, no effect at all was supposed to occur. However,

besides coherent correlations, some results were paradoxical and remained incomprehensible in a classical

framework. However, using a quantum-like model, the probabilities of the different outcomes could be calculated

according to the different experimental contexts. Conclusion: In this reassessment of an historical series of

“memory of water” experiments, quantum-like probabilities allowed modeling these controversial experiments

that remained unexplained in a classical frame and no logical paradox persisted. All the features of Benveniste’s

experiments were taken into account with this model, which did not involve the hypothesis of “memory of water”

or any other “local” explanation.

Key Words:

memory of water, quantum-like probabilities, quantum cognition, entanglement, contextuality, non-

local interactions

NeuroQuantology 2013; 2: 197-208

“One explanation might be that the data had been

generated by a hoaxer in [Benveniste’s]

laboratory.” (Maddox 1988)

Introduction

1

Some words – such as “memory of water” –

have the remarkable property to induce rapid

and strong physiological reactions in readers,

especially if they are also science editors. No

doubt that classical Pavlovian conditioning is

Corresponding author: Author name

Address: Francis Beauvais, MD, PhD. 91, Grande Rue, 92310 Sèvres,

France.

Phone: + 33 1 45 34 92 20

Fax: +33 1 79 72 31 60

beauvais@netcourrier.com

Received April 9, 2013. Revised April 15, 2012.

Accepted April 25, 2012.

eISSN 1303-5150

at work (Reiff et al., 1999). The present article

should not induce any hypertensive response

since I will describe a series of Benveniste’s

experiments without reference to modification

of water structure whatsoever. Indeed, I

proposed recently to model these controversial

experiments with some notions inspired from

the generalized probability theory that is the

core of quantum physics (Beauvais, 2012;

2013). Strictly speaking, the possibility of

“memory of water” was not definitely

dismissed; it is always difficult to prove that

something does not exist. Nevertheless, all

difficulties encountered by Benveniste

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198

(reproducibility, disturbances after blinding)

were described in this quantum-like model,

which did not require the hypothesis that

water had been “structured” or “informed”.

The controversy with the journal

Nature and its editor has been extensively

discussed (de Pracontal, 1990; Schiff, 1998;

Benveniste, 2005; Beauvais, 2007; Thomas,

2007; Beauvais, 2012). The above quote of J.

Maddox, the editor of Nature during the

“Benveniste’s affair”, is a good indicator of the

state of mind of some scientists faced with the

puzzling results on the effects of high dilutions

reported in the Nature’s article (Davenas et

al., 1988). Less known are the experiments

performed by Benveniste’s team after

publication of the controversial article in 1988.

Thus, a large series of blind experiments with

the same basophil model was performed under

the supervision of statisticians and statistically

significant results were obtained in favor of the

effects of high dilutions confirming both the

results of 1988 in Nature and other results

previously published with the same biological

model (Davenas et al., 1987; Benveniste et al.,

1991). Nevertheless, Benveniste abandoned

basophils and searched for other models that

were less disputed.

One of the biological systems that were

routinely used in Benveniste’s laboratory –

namely the isolated perfused

rodent heart preparation (Langendorff

preparation) – was shown to respond to high

dilutions of various pharmacological

compounds (Hadji et al., 1991; Benveniste et

al., 1992). The Langendorff heart preparation

is a classical model of physiology, which allows

recording pharmacological effects of biological

compounds or pharmacological drugs on

different parameters of a rodent heart

maintained alive. In early experiments with

high dilutions, coronary flow appeared to be

the most sensitive parameter. This biological

model had the advantage to be more objective

than basophil counting, which depends on the

judgment and skill of the experimenter.

Moreover, with the Langendorff preparation,

the biological effects of high dilutions were

directly observed in the series of tubes that

collected the effluent from coronary arteries.

Therefore, in contrast with the basophil model,

the effects of high dilutions could be shown in

real time to scientists interested by this

research who visited the laboratory.

In 1992, Benveniste reported that he

was able to transmit the “molecular

information” contained in an aqueous solution

by placing a tube containing a biologically

active compound in an electric coil at the entry

of a low-frequency amplifier; the “biological

information” was said to be transmitted to

naïve water contained in another tube placed

in a second electric coil wired at the amplifier

output. In a further refinement (1995), the

“molecular signal” was digitized and stored on

the hard disk of a personal computer and

could then be “replayed” in a second time to

naïve water. Benveniste coined then the term

“digital biology”. In the last version (1997), the

coil was directly fixed on the perfusion column

of the Langendorff system and therefore the

system could be piloted from the computer

without injection of the samples of “informed”

water into the perfusion circuitry. The results

obtained with these successive devices were

published as posters and abstracts at

congresses (Aïssa et al., 1993; Benveniste et

al., 1994; Aïssa et al., 1995; Benveniste et al.,

1996; Benveniste et al., 1997; Benveniste et al.,

1998). If true, these “discoveries” were

ground-breaking, but they received great

skepticism (Schiff, 1998; Beauvais, 2007).

In order to convince other scientists

that his controversial research was well-

founded, Benveniste organized regular public

demonstrations during years 1992–1998.

During these demonstrations, experimental

samples were produced and blinded by

participants (Beauvais, 2007). The samples

were then assessed on the Langendorff system.

The initial objective of these demonstrations

has however never been achieved because an

unexpected phenomenon occurred repeatedly.

Indeed, after unblinding of the masked

experiments, a “signal” was frequently found

with “control” tubes whereas some samples

supposed to be “active” were without effect.

Benveniste generally interpreted these failures

as “jumps of activity” between samples and as

a logical consequence he concluded that

“informed” water samples should be protected

from external influences, particularly

electromagnetic waves. Despite additional

precautions and further improvements of the

devices, this weirdness nevertheless persisted

and was an obstacle for the establishment of a

definitive proof of concept (Benveniste, 2005;

Beauvais, 2007; Thomas, 2007; Beauvais,

2008; 2012).

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The purpose of the present article is

first to describe in detail such a demonstration

that comprised a series of experiments made

in parallel; these experiments were blinded

and closely controlled by observers not

belonging to Benveniste’s team. The aim of

these experiments was to demonstrate

“electromagnetic transmission of biological

activity” to naïve water. In a second time, we

will see how the results of these experiments

that remain inexplicable in a classical

framework, are easily described in a quantum-

like model without reference to “memory of

water”.

Methods

The protocol of the experiments of May

13th

1993

The public demonstration described in this

article included four parallel independent

blind experiments starting on May 13

th

1993.

For this purpose, a written protocol precisely

described the experiments and defined the role

of each participant. After completion of all

measurements, the raw data were presented to

the participants before unblinding. An internal

report reported the results and included all

data and original records.

This series of experiment was designed

and proposed to Benveniste by Michel Schiff

who was a former physicist who turned next to

psychology and social sciences (Schiff et al.,

1978). He was amazed by the “memory of

water” controversy and had no a priori

opinion on the debate on “memory of water”.

Schiff proposed to Benveniste to spend time in

his laboratory to get information on this

research; in exchange he could bring help,

particularly for design of experiments,

statistical analysis and supervision of

experiments. Schiff joined Benveniste’s

laboratory half-time during years 1992–1993;

he reported his experience in a book (Schiff,

1998). The purpose of the demonstration was

to convince the participants that it was

possible, according to the title of the internal

report, “to dissociate molecular information

from its support and to transmit it to naïve

water”.

The electronic devices have been

described in details elsewhere (Thomas et al.,

2000); it was composed of a low-frequency

amplifier with a coil wired at input (for

pharmacological solution) and a coil wired at

output (for “imprinting” of naïve water). The

biological model was the Langendorff

preparation, which allows maintaining alive a

rodent heart while pharmacological agents are

injected into the circuitry to modify some

physiological parameters (Beauvais, 2007;

2012). Change of coronary flow was the main

biological parameter that was recorded with

this system in Benveniste’s experiments on

“memory of water”.

On May 13

th

1993, the participants to

these experiments met in a laboratory at Paris.

Four parallel experiments of “electronic

transmission” were performed by four teams;

each team was composed of two participants

who were not members of Benveniste’s

laboratory. An original method for blinding of

sample labels was used so that nobody knew

the original label until unblinding. The

molecule to be transmitted was ovalbumin and

rats of which hearts were used for

measurements had been sensitized to

ovalbumin.

The successive tasks of each two-

participant team (one participant performed

the experimental handlings and the other was

a witness) were the following: choice of ten

plastic tubes containing distilled water from a

stock and choice of ten padded envelopes from

a stock; one tube was placed in each envelope.

One envelope was chosen and the respective

tube was placed on the output coil of the low-

frequency amplifier (a tube containing

ovalbumin at 10 µmol/L was always present on

the input coil). After 15 min, the “transmitted”

tube was placed again into the envelope with a

self-adhesive label attached inside the

envelope. The nine other tubes of naïve water

were left untouched and the ten envelopes

were mixed for randomization. Then all tube

received labels with code: each tube was

extracted from envelope (without looking

inside), received a self-adhesive label and an

identical label was placed on the envelope

(outside); the labels were 1 to 10 for experience

#1, 11 to 20 for experience #2; 21 to 30 for

experience #3 and 31 to 40 for experience #4.

All envelopes were given to a bailiff who kept

them until unblinding. Before and after each

“ovalbumin transmission”, one open-label

transmission was also performed by a member

of Benveniste’s team (positive controls).

The 40 blinded tubes and the 8 open-

label positive controls were then transported

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to Benveniste’s laboratory at Clamart, in the

inner suburbs of Paris. The content of all

samples was assessed from May 13 to 17 on the

Langendorff device: for homogeneity of

results, each series was tested on the same

heart (one heart per series). For each of the

four experiments, one open-label water sample

(negative control), one open-label sample of

“transmitted” ovalbumin (positive control)

and the ten blinded samples were assessed;

the last sample was a sample of ovalbumin at

0.1 µmol/L (positive control at “classical”

concentration).

After a first measurement of all

samples, the tubes received a new code and

another round of measurements was

undertaken. This interim blinding was

performed by Schiff and another member of

Benveniste’s laboratory not involved in these

experiments.

The quantum formalism in brief

In quantum physics, all the knowledge on a

physical object is summarized by a state vector

. For a system S with two possible states S

1

and S

2

(e.g., disintegrated and non-

disintegrated states of a radioactive atom), the

state of the system S is described by the

following state vector:

1 2S

a S b S

This equation means that before

measurement the quantum object is in a

“superposed” state described by the sum of

two state vectors. It is important to note that

the indetermination of the state of the system

before measurement is total (there is no

“hidden variables”). After measurement

(“reduction of the quantum wave”), the

probability P1 to observe S

1

is a

2

and the

probability P2 to observe S

2

is b

2

.

In classical probability theory,

probabilities add. Thus, if P1 and P2 are the

probabilities associated to two mutually

exclusive events S

1

and S

2

, the probabilities for

either event to occur is Prob (S

1

or S

2

) = P1 +

P2. In contrast, in quantum probability theory,

probability amplitudes add and probabilities

are calculated as the square of probability

amplitudes. Thus, if a and b are the probability

amplitude associated to two events S

1

and S

2

(with P1 = a

2

and P2 = b

2

), then:

Prob (S

1

or S

2

) = (a + b)

2

= P1 + P2 +

interference term.

The interference term is added or

subtracted to classical probabilities according

to sign to give quantum probabilities.

The notion of non-commutable

observables is another key concept of quantum

probabilities. Physical “observables” are

mathematical “operators” and for each

operator there is a spectrum of possible

results, which are named the “eigenvectors” of

the operator (they constitute an orthogonal

basis in the vector space). When an operator is

applied to a state vector, the vector is split into

different components, which are the

eigenvectors of the operator (Figure 1). If the

original state vector to be observed is an

eigenvector of the operator, then it is not

affected (this means that the value of the

parameter to be measured was already fixed

before measurement). Two observables are

said to commute with each other when they

share eigenvectors (the shared eigenvectors

are not affected by the measure of the other

observable). As a consequence, the outcomes

will be different according to the order of the

measurements. When two observables are not

commutable, the set of eigenvectors of one

observable (orthogonal basis) can be expressed

as a linear combination of the set of

eigenvectors of the other observable; in other

words, there are two different bases for the

same vector space.

Type-1 and type-2 observers in

quantum-like model

The point of view of the different types of

participants/observers must be precisely

defined. We will now refer to an “inside”

observer as a type-2 observer and an “outside”

observer as a type-1 observer.

The emphasis placed on the different

points of view of observers is reminiscent of

the thought experiment named “Wigner’s

friend” proposed in the early 1960s by the

physicist Eugene Wigner (Figure 2)

(D'Espagnat, 2005; Wikipedia, 2013).

Actually, “Wigner’s friend” was an extension of

another famous thought experiment, namely

Schrödinger’s cat. Wigner’s friend is supposed

to perform a measurement on a macroscopic

system (Schrödinger’s cat) linked to a

microscopic quantum system (radioactive

atom), which is in a superposed state before

measurement. Wigner remains outside the

laboratory and he has no information on the

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state of his friend. At the end of the

experiment, from the point of view of Wigner’s

friend, the cat is either dead or alive

(“collapse” of the quantum wave from a

superposed state). From the point of view of

Wigner, the cat is in a superposed state of the

two possible outcomes: cat dead and alive

(with Wigner’s friend in the corresponding

state). If Wigner enters the laboratory or has

information on the result of the experiment

(“collapse” of the quantum wave), he learns

that the cat is dead or alive and his friend is in

the corresponding state. This is the

“measurement problem”: we have two valid

but different descriptions of the reality with

apparent “collapse” of the quantum wave at

different times according to the different

observers.

Figure 1. Design of an experiment exhibiting quantum-like interferences (application to Benveniste’s experiments). The

quantum object (cognitive state A of the experimenter) is symbolized by the state vector

A

ψ

and is measured through two

successive observables, which are mathematical operators. The first observable (“labels”) splits the state

into two

orthogonal states (denoted

IN

A

and

AC

A

). Each of these two states is split by the second observable (“concordance of

pairs”) into two new orthogonal states,

CP

A

and

DP

A

. It is assumed that the observables do not commute. If the events

inside the box are not measured/observed, the system is in a superposition of states, which is not equal to a mixture of the two

states. The consequence of superposition is that quantum probabilities to observe

CP

A

or

DP

A

are different compared

with classic probability. Indeed, quantum probabilities (P

II

) are calculated as the square of sum of probability amplitudes of

paths; classical probabilities (P

I

) are calculated as the sum of squares of probability amplitudes of paths.

In Benveniste’s experiments, the type-1

observer (“outside”) is the equivalent of

Wigner whereas type-2 observer (“inside”) is

the equivalent of Wigner’s friend (Figure 2).

The type-2 observer is on the same “branch of

reality” of the experimenter with experimental

device (i.e., Schrödinger cat); the type-1

observer considers that the type-2 observer (or

the experimenter) is in a superposed state

(until he interacts with him).

Statistical analysis

The raw data were obtained from the internal

report of Schiff and Benveniste and the

analysis of the results was reassessed. The

biological parameter that was recorded during

these experiments was the coronary flow

recorded for 15 min (one time point per min).

When a signal was recorded, the flow change

was maximal at 3–4 min after injection of

sample into perfusion circuitry and flow

returned to basal value before 10 min. The

area under the curve (AUC) method was used

to present the results in this article. The mean

and standard deviations of background were

calculated with the nine samples (in each of

the four experiments) that did not significantly

change the coronary flow. The experimental

result obtained with each sample was

expressed as the number of standard

deviations of background from mean

background. Another method was used to

summarize the results in the internal report of

Schiff and Benveniste; the results were

sufficiently clear-cut to lead to identical

conclusions for identification of samples

associated to signal and background in each

experiment.

Observable #1

Observable #2

Observable #2

A

ψ

CP

A

DP

A

IN

A

(a)

(b)

AC

A

P

II

(A

CP

) = |a cos θ + b sin θ|

2

P

I

(A

CP

) = a

2

cos

2

θ + b

2

sin

2

θ

(µ

11

= cos θ)

(µ

12

= –sin θ)

(µ

22

= cos θ)

(µ

21

= sin θ)

P

II

(A

DP

) = |b cos θ – a sin θ|

2

P

I

(A

DP

) = b

2

cos

2

θ + a

2

sin

2

θ

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Figure 2. Type-1 observer (Wigner) and type-2 observer

(Wigner’s friend). In this thought experiment, two points of

view are successively considered. From the point of view of

Wigner who has no information on experiment outcome, the

chain of measurement including his friend is in an

undetermined state at the end of the experiment (superposed

state). There is “collapse of the quantum wave” when Wigner

enters the laboratory and learns the outcome of the

experiment. From the point of view of Wigner’s friend,

“collapse” occurs when he looks at the measurement

apparatus at the end of the experiment and he never feels

himself in a superposed state; on the contrary he feels that

one of the outcomes has occurred with certainty. Therefore

two valid but different descriptions of the reality coexist in

this thought experiment with apparent “collapse” of the

quantum wave at different times according to information

that observers get on quantum system. In Benveniste’s

experiment, we make a parallel with Wigner (type-1 observer)

and Wigner’s friend (type-2 observer) to define the point of

view of the different participants/observers.

Results

Results of the four experiments and

interpretation by Benveniste’s team

The results of the two rounds of measurements

in the four parallel experiments are described

in Table 1 and Table 2. As “expected”, a signal

corresponding to one sample and only one

emerged from background in each 10-samples

series in first round of measurements: label #8

in first experience, label #17 in second

experience; label #21 in third experience and

label #34 in fourth experience. This is not a

trivial comment since, according to

mainstream science, no effect at all was

supposed to occur.

Table 1. Results of the Benveniste’s experiments of May 13

th

, 1993: first round of measurements after blinding by type-1

observers.

Exp. #1 Exp. #2 Exp. #3 Exp. #4

Label # Result Label # Result Label # Result Label # Result

Blind samples: in each series, 9 “inactive” labels (water) and 1 “active” label (Ova. tr.)

1 -0.5 11 1.5 21 13.2 31 0.6

2 -1.2 12 -1.3 22 -0.5 32 0.6

3 -0.8 13 -0.6 23 -1.0 33 0.6

4 0.2 14 -0.6 24 -0.5 34 10.8

5 2.0 15 -0.6 25 0.1 35 0.6

6 -0.8 16 -0.6 26 -1.0 36 -0.9

7 -0.1 17 21.4 27 0.6 37 1.0

8 16.0 18 0.8 28 2.2 38 0.6

9 0.6 19 1.5 29 -0.5 39 -1.7

10 0.6 20 0.1 30 0.6 40 -1.3

Open-label samples

Water

a

-0.5 Water -1.3 Water 0.6 Water -0.2

Ova. tr.

b

8.1 Ova. tr. 9.0 Ova. tr. 19.5 Ova. Tr. 14.5

Ova.

c

43.7 Ova. 37.2 Ova. 27.9 Ova. 20.2

Results are expressed as the number of standard deviations of background from mean background (see Methods section).

Results corresponding to “emergent signal” are in bold characters in grey boxes.

a

Negative control of water (no “transmission”)

b

Positive control: water “informed” with ovalbumin (Ova.) “transmitted” (tr.) through electronic device

c

Positive control: ovalbumin at “classical” concentration (0.1 µmol/L).

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After the second round of

measurements, again a signal corresponding

to one sample and only one emerged from

background in each experiment. As indicated

in Table 2, the four samples that were

associated with signal for the second round

were the same than for the first round despite

interim blinding. This was an important result

for Benveniste’s team, since it strongly

suggested that all series were successful.

Table 2. Results of the Benveniste’s experiments of May 13

th

, 1993: second round of measurement after interim blinding by

type-2 observers (in-house blinding).

Exp. #A (Exp. #1)

Exp. #B (Exp. #3) Exp. #C (Exp. #4) Exp. #D (Exp. #2)

Label # Result Label # Result Label # Result Label # Result

Blind samples: in each series, 9 “inactive” labels (water) and 1 “active” label (Ova. tr.)

A (6)

a

- B (30) 1.8 D (32) 0.3 C (14) 1.6

E (8) 25.1 F (25) 0.0 H (31) 1.1 G (11) 1.6

O (3) 1.3 N (27) 0.0 J (35) 1.1 I (16) 0.4

Q (2) -1.3 P (21) 11.6 M (38) -0.5 K (13) -0.5

U (4) 0.0 W (28) 0.0 S (39) 1.1 L (18) 0.0

V (7) -1.3

AB (29) -0.9 T (40) -1.4 R (19) -0.9

AA (1) 1.3 AG (26) -1.8 Z (33) -1.4 X (15) -0.9

AD (9) 0.0 AH (22) 0.9 AE (36) -0.5 Y (20) -0.9

AF (5) 0.0 AI (24) 0.0 AK (34) 11.7 AC (17) 4.1

AM (10) 0.0 AJ (23) 0.0 AN (37) 0.3 AL (12) -0.5

Open-label samples

Water

b

0.0 Water 2.7 Water 1.1 Water 0.0

Ova. tr.

c

9.3 Ova. tr. 8.9 Ova. tr. 19.1 Ova. tr. 6.6

Ova.

d

- Ova. 20.6 Ova. 23.1 Ova. 7.4

Results are expressed as the number of standard deviations of background from mean background (see Methods section).

Results corresponding to “emergent signal” are in bold characters in grey boxes.

a

Number between parentheses is label # from Table 1.

b

Negative control of water (no “transmission”)

c

Positive control: water “informed” with ovalbumin (Ova.) “transmitted” (tr.) through electronic device

d

Positive control: ovalbumin at “classical” concentration (0.1 µmol/L).

On May 19, all participants had a

meeting at the same location as previously in

Paris to assist to the unblinding of the

experiments. Results were first presented and

then envelopes were opened by the bailiff for

unblinding. Labels that were “expected” to be

associated with signal were revealed: #8, #18,

#26 and #34. Therefore, experiments #1 and

#4 were successes and experiments #2 and #3

missed the target.

These results were considered as

illogical by Benveniste’s team. Indeed, this was

a half-success: two experiments had signal at

the expected place; but why the target was

missed in the two series of measurements

despite coherent results after interim in-house

blinding was baffling. Schiff calculated the

probabilities of different scenarios supposing

an experimental artifact (internal report). The

first hypothesis was that the artifact was

located in the measurement device

(Langendorff preparation) supposing a

discontinuous functioning in an all-or-nothing

manner. The second hypothesis was that

contamination of some tubes would be

responsible of all-or-nothing effects. In both

cases (random false positive results or random

contamination), the probabilities were very

low and these hypotheses were rejected. Schiff

concluded that only trivial errors (such as label

mistakes between transport of tubes after

blinding and first measurement) could explain

these weird results. No objective data however

supported this conclusion. It is important to

note that these hypothetical scenarios rested

on the assumption that “something” was

present in water samples. This is what could

be named a “local” interpretation.

Benveniste concluded that the

experimental devices needed to be improved

and he continued his endless technical pursuit

for the decisive experiment. Among other

improvements, he developed what he named

“digital biology” to reduce the possible

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contaminations or electromagnetic

interferences. However, the spontaneous

“jumps” of activity between “active” and

“inactive” samples and other weirdness

persisted (Beauvais 2007).

In the next parts of the text, we will

describe these experiments using quantum-

like probabilities.

The quantum-like formalism applied to

Benveniste’s experiments

Definitions

The purpose of the experiments performed by

Benveniste was to assess the rate of

concordant pairs, namely “inactive” samples

(IN) with background noise (“↓”) and “active”

samples (AC) with signal (“↑”). In other words,

we must quantify the correlation between

“expected” results and observed results.

We define P

I

(A

CP

) as the probability for

the cognitive state (named A) of the

experimenter to be associated with concordant

pairs (CP) according to classical probabilities;

P

II

(A

CP

) is the same probability according to

quantum probabilities. P

I

(A

DP

) and P

II

(A

DP

)

are the respective P

I

(classical) and P

II

(quantum) probabilities for discordant pairs

(DP).

We describe the experimental situation

from the point of view of an external observer

that knows the initial state of the system and

does not perform any measurement /

observation.

Open-label or type-2 blinding

The state vector of the cognitive state A of the

experimenter is described in terms of the

eigenvectors of the first observable (cognitive

states A indexed with labels IN and AC):

A IN AC

a A b A

for each sample in

each series.

The probabilities a

2

and b

2

associated

with the states A

IN

and A

AC

are the proportions

of samples with IN and AC labels, respectively.

We develop the eigenvectors of the first

observable on the eigenvectors of the second

observable (concordance of pairs). We

postulate that the cognitive states A indexed

with “labels” and the cognitive states A

indexed with “concordance of pairs” are non-

commutable observables:

11 12IN CP DP

AAA

21 22AC CP DP

A A A

Therefore, we can express

A

as a

superposed state of

CP

A

and

DP

A

:

11 21 12 22

( ) ( )

A CP DP

a b A a b A

The probability of A

CP

is the square of the

probability amplitude associated with its state:

2

11 21

( )

II CP

P A a b

Type-1 blinding

If a type-1 observer has blinded the labels, the

context of the experiment changes. In this

case, one observable (labels) is measured

/observed by the type-1 observer and not by

the experimenter as above; this is formally

equivalent to a which-path measurement in

single-particle experiment. The cognitive state

A cannot interfere with itself (there is no

superposition of

IN

A

and

AC

A

) and classical

probabilities apply for calculation of the

probability of concordant pairs (Figure 1):

( ) ( ) ( | ) ( ) ( | )

I CP IN CP IN AC CP AC

P A P A P A A P A P A A

With

2

11

(|)

CP IN

P A A

and

2

21

( | )

CP AC

P A A

,

then:

2 2

2 2

11 21

( )

I CP

P A a b

Similarly,

2 2

2 2

12 22

( )

I DP

P A a b

.

We note that, in the general case, the

probability for A to be associated with

concordant pairs is dependent on the

experimental context (open-label/type-2

blinding vs. type-1 blinding) since we find

P

I

(A

CP

) ≠ P

II

(A

CP

). The difference is due to the

interference term.

Simplification of the formalism equations

Since

1

2

12

2

11

,

1

2

22

2

21

and

1)()(

DPIICPII

APAP , we can easily calculate

that

11 21 22 12

,

2 2

11 22

and

2

21

2

12

.

Thus, we can write:

11 21IN CP DP

AAA

21 11AC CP DP

A A A

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205

We note that the matrix for change of basis is a

rotation matrix; counterclockwise rotation has

been chosen for appropriate concordance

(experimenter’s choice) between labels

(IN/AC) and biological outcomes

(background/signal):

11 12 11 21

21 22 21 11

cos sin

sin cos

Therefore,

( cos sin ) ( cos sin )

A CP DP

a b A b a A

The formulas of P

II

(A

CP

) and P

I

(A

CP

) become

(Table 3 and Figure 1):

P

II

(A

CP

) = |a cos θ + b sin θ|

2

P

I

(A

CP

) = a

2

cos

2

θ + b

2

sin

2

θ

with

P

I

(A

CP

|A

IN

) = cos

2

θ and P

I

(A

CP

|A

AC

) = sin

2

θ

The formulas of P

II

(A

DP

) and P

I

(A

DP

) are

similarly calculated:

P

II

(A

DP

) = |b cos θ – a sin θ|

2

P

I

(A

DP

) = b

2

cos

2

θ + a sin

2

θ

with

P

I

(A

DP

|A

IN

) = sin

2

θ and P

I

(A

DP

|A

AC

) = cos

2

θ

In a previous paper, this model allowed

describing Benveniste’s experiments without

any reference to “memory of water”,

“electronic transmission”, “digital biology” or

any other “local” explanation (Beauvais, 2013).

Just supposing superposed states and non-

commutable observables, the quantum-like

model described the main characteristics of

Benveniste’s experiments: emergence of signal

from background, different outcomes

according to type-1 or type-2 blinding and

apparent “jumps of activity” between samples.

We remind briefly these issues using the

quantum-like model.

Emergence of signal from background

If θ = 0, then the observables are commutable:

cos sin

1 0

IN CP DP

CP DP CP

A A A

A A A

sin cos

0 1

AC CP DP

CP DP DP

A A A

A A A

In this case, the two observables share their

eigenvectors:

IN CP

A A

and

AC DP

A A

.

The observation of concordant pairs is always

associated with label IN (i.e., IN always

associated with “↓”) and the observation of

discordant pairs is always associated with label

AC (i.e., AC always associated with “↓”). In

other words, no signal is observed when the

observables are commutable (θ = 0) since only

background is associated with both IN and AC

labels. Therefore, non-commutable

observables are necessary for signal

emergence. The signal must be one of the

possible states of the system, even with a low

probability. Thanks to entanglement, the

probability of signal increases. In a previous

article, we proposed that the relationship

between different cognitive states (A

IN

with A

↓

and A

AC

with A

↑

), which are summarized in θ

value, results of associative processes related

to cognition mechanisms (Beauvais, 2013).

Table 3. Summary of the quantum-like model describing Benveniste’s experiments.

Non-commutable observables (θ ≠ 0) Commutable

observables

(θ = 0)

With

interference term

(superposition)

Without

interference term

(no superposition)

Presence of signal Yes

a

Yes

b

No

c

Concordance of labels and outcomes

d

High

e

Low NA

Probability of concordant pairs: P(A

CP

) |a cos θ + b sin θ|

2

a

2

cos

2

θ + b

2

sin

2

θ a

2

Probability of discordant pairs: P(A

DP

) |b cos θ – a sin θ|

2

b

2

cos

2

θ + a

2

sin

2

θ b

2

Corresponding experimental situations

Open-label or

blinding by

type-2 observer

Blinding by

type-1 observer

Unqualified or

untrained experimenter

NA: not applicable

a

P

II

(A

↑

) = a

2

× P

II

(A

DP

) + b

2

× P

II

(A

CP

); a

2

is the proportion of “inactive” labels (IN) and b

2

is the proportion of “active” labels (AC)

b

P

I

(A

↑

) = sin

2

θ

c

Observables are commutable with cos θ = 1 and sin θ = 0; then P(A

↑

) = 0 and P(A

↓

) = 1 (only background is associated with A;

there is no signal)

d

Concordant pairs : A

IN

associated with A

↓

or A

AC

associated with A

↑

e

For sin θ = b (and consequently cos θ = a), the quantum interference is maximal with

P

II

(A

CP

) = 1 and P

II

(A

DP

) = 0.

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206

Outcomes after type-1 blinding or type-

2 blinding

We note first that open-label experiments or

experiment after blinding with type-2 observer

are not formally different since experimenter A

and type-2 observer O are on the same

“branch” of the reality described by the state

vector (Figure 1) (Beauvais 2013):

( cos sin )

( cos sin )

AO CP CP

DP CP

a b A O

b a A O

With the above formulas, we calculate now

the outcomes of experiments by supposing

that the number of “inactive” samples (labels

IN) and “active” samples (labels AC) are equal

(a

2

= 0.5 and b

2

= 0.5); we suppose that

quantum-like correlations are optimal

(cos θ = a and sin θ = b):

P

II

(A

CP

) = |a cos θ + b sin θ|

2

= 1

P

II

(A

DP

) = |b cos θ – a sin θ|

2

= 0

P

I

(A

CP

|A

IN

) = cos

2

θ = 0.5

P

I

(A

CP

|A

AC

) = sin

2

θ = 0.5

Therefore, after blinding with type-2

observer (or in open-label experiments), all

samples with IN labels are associated with

background and all samples with AC label are

associated with signal. In contrast, after

blinding with type-1 observer, P

I

(A

CP

) = 0.5

and P

I

(A

DP

) = 0.5. In other words, in type-1

blind setting, the proportion of samples with

AC labels associated with signal decreases

from 100% to 50% and the proportion of

samples with IN labels associated with signal

increases from 0% to 50%. Therefore,

everything happens as if “biological activity”

(signal) “jumped” from some samples with AC

label to samples with IN label. These apparent

“jumps” of activity between samples were

precisely a blocking issue in the

demonstrations aimed to provide a proof of

concept on the reality of the biological effects

related to “memory of water”. Therefore, our

quantum-like model easily describes these

“disturbances” without supposing additional

hypotheses involving “external” causes or

experimental artifacts.

Numerical application

We are now able to apply these calculations to

the historical series of Benveniste’s

experiments described in this article; in each

series, one unique sample with “active” label

had to be “guessed” out of ten (i.e., a

2

= 0.9

and b

2

= 0.1). With the open-label samples or

after type-2 blinding, the probability of

concordant pairs was maximal; therefore, for

all experiments (including type-1

experiments), we take sin θ = b.

In experiments of May 13

th

1993, two

“successes” out of four (50%) were observed;

the 95% confidence interval of this proportion

is [0.068–0.932] (Clopper-Pearson confidence

interval for a binomial parameter).

According to the formalism, after type-

1 blinding, the probability for a sample

(regardless label) to be associated with signal

is random and is therefore b

2

= 0.1; among

series of ten samples, the probability to draw a

series with one and only one signal is 0.29

(binomial law). Therefore, the theoretical

probability to “draw” the “good” sequence (a

10-sample series with signal at the same place

as AC label) is 0.29 × 0.1 = 0.029; this value is

excluded of the 95% confidence interval

calculated above. However, we must consider

that some parts of any blind experiment are

nevertheless open-label: in the present case,

one active sample and nine inactive samples in

each series was defined by the protocol and

was available information. Therefore, statistics

must be applied on the subgroup of

permutations of ten samples with one and only

one signal. The theoretical probability for

“success” (one unique signal at the expected

place) is then 0.1 (and not 0.029 as above).

This value is now included in the calculated

95% confidence interval.

More than accuracy of calculation, the

important point is that, after type-1 blinding,

probability for “success” is strongly decreased.

Moreover, taking into account all information

available to the experimenter allows better

fitting with the quantum-like model.

Discussion

The “public demonstration” of Benveniste’s

experiments described in the present article

was performed with a wealth of precautions

rarely achieved in usual research. Many

witnesses were involved and in-house blinding

was superimposed to blinding by “outside”

participants. It is important to emphasize

again that a signal was found associated to

four samples out of forty; this result was

important and remains unexplained in the

present knowledge of science. However, only

two signals were at the “expected” place.

Therefore, the demonstration was a “half-

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207

success”; totally convincing results were

paradoxically not achieved although the test of

in-house blinding was passed with complete

success.

The main problem in the “memory of

water” experiments is not so much the lack of

explanation on the origin of these phenomena,

but the absence of a logical framework.

Indeed, faced with the results of Benveniste’s

experiments, there is an unavoidable dilemma

if we interpret them in a classical frame.

Indeed, if we assume – as Benveniste did –

that “something” was present in samples with

“active” labels (hypothesis of “memory of

water”), we are then unable to explain why

these experiments failed more frequently than

expected after type-1 blinding (two out of four

experiments in the data presented in this

article). If we tempt to explain the “jumps of

activity” as artifacts (random triggering of the

measurement apparatus or random

contamination), probability calculations do

not support such hypothesis. Of course, we can

also tempt to dismiss the hypothesis of

“memory of water” and its avatars, but we are

unable to explain the emergence of a signal

from background and a bulk of coherent

results (such as the significant correlations

that persisted after type-2 blinding).

A third possibility is to change the

logical framework and to use a generalized

probability theory (that includes classical

probability theory as a limit theory). In this

later case, emergence of signal and

presence/absence of correlations according to

experimental context are simply described

without additional ad hoc hypotheses. The

passage from classical to quantum logic

requires only non-null value for the parameter

θ. The alternate way proposed by quantum-

like formalism is obviously not intuitive.

Nevertheless, if we accept an effort of

abstraction, a couple of simple equations can

give a formal framework to these poorly

understood experiments and quantitative

statistical modeling can be performed.

In this quantum-like framework, there

is no paradox; “successes” and “failures”

appear then as the two faces of the same coin.

In the paradigmatic two-slit experiment of

Young, observing “waves” (interference

pattern on the screen) is not considered as a

success whereas observing “particles” (no

interference pattern after which-path

measurement) is not considered as a failure. In

the quantum-like model of Benveniste’s

experiments, we can decide to observe either

“waves” (high rate of correlated pairs in open-

label or type-2 blind settings) or “particles”

(low rate of correlated pairs in type-1 blind

setting) (Beauvais 2013). “Waves” and

“particles” are two complementary aspects of

the same quantum (or quantum-like) object.

Table 4 summarizes the “successes” and

“failures” of Benveniste’s experiments

according to the different experimental

contexts.

Table 4. Contextuality in Benveniste’s experiments: three different patterns of results are observed according to experimental

context.

Experimental context

Open-label or blinding

by type-2 observer Blinding by type-1

observer

Unqualified or

untrained

experimenter

Expected results

a

↓↓↓↓↑↑↑↑ ↓↓↓↓↑↑↑↑ ↓↓↓↓↑↑↑↑

Observed results ↓↓↓↓↑↑↑↑ ↓↓↑↑↓↑↓↑ ↓↓↓↓↓↓↓↓

Probability of concordant pairs 1 1/2 1/2

Description of results Signal present at

expected places

Signal present but at

random places No signal

Conclusion according

to classic logic Success Failure

(“jumps of activity”

between samples)

Failure

Conclusion according

to quantum logic

θ ≠ 0 with

superposition of

quantum states

(interferences)

θ ≠ 0 without

superposition of

quantum states

(no interferences)

θ = 0 (classical

probabilities apply)

a

Experiments with equal number numbers of “inactive” and “active” labels and with maximal quantum interferences (a

2

= b

2

= 0.5 and sin θ = b).

NeuroQuantology | June 2013 | Volume 11 | Issue 2 | Page 197-208

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208

This quantum-like model is in the spirit

of quantum cognition, an emerging research

field that proposes to model cognitive

mechanisms and information processing in

human brain by using some notions from the

formalism of quantum physics such as

contextuality or entanglement. Using

quantum-like probabilities allowed addressing

problems that appeared paradoxical in a

classical frame. These new tools have been

applied to human memory, decision making,

personality psychology, etc (see for example

the special issue of Journal of Mathematical

Psychology in 2009) (Bruza et al., 2009).

Conclusions

The “paradoxical” results of a series of

Benveniste’s experiments performed in 1993,

which were closely controlled and blinded by

observers not belonging to Benveniste’s team,

were reassessed. Using a quantum-like model,

the probabilities of the different outcomes

were calculated according to experimental

context and no logical paradox persisted. All

the features of Benveniste’s experiments were

taken into account with this model, which did

not involve the hypothesis of “memory of

water” or any other “local” explanation.

Acknowledgements

This article is dedicated to the late Jacques

Benveniste and Michel Schiff.

Author disclosure statement

No conflict of interest.

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